By "changing the core" in the solenoid, I believe you mean changing the material used.
If it's a DC solenoid (and most are), the difference will only occur when the current changes - when it is first applied and when it is first removed. But during those transition periods, the current will also be affected by the position of the solenoid arm. As the arm moves in, the inductance rises - slowing the rise of the current. Since the arm may be moving against a load, the actual current change during transition will be effected by the magnetic permeability of the arm as well as the mechanical force it generates.
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